## https://arxiv.org/pdf/1504.04885.pdf

Quaternion formalism and physics, 1880-1940

## Averaging

Does the action of the averaging operator factorize the dimension of the wheel in to the substances of the wheel writings?

## Colossus c(l) osets vs statistics

It’s now clear to me that while good et al thought of the attack on tunny in purely statistical theory, Newman and Turing thought of it in terms of topology (and log linear multinomials)

We have yet to see a Newman/Turing analysis (or topology applied to either enigma or tunny (or Italian heybern)) yet.

I see why it would still be seen as sensitive.

## Metadata on trump

When I wrote my last memo I didn’t know that it was part of a wider maelstrom.

It’s clear that the Truisms of all American (cum British) lying are abound; with definitional lying at the fore.

In my Era, pre 911, via 2 layers of Anglo-American (unofficially well coordinated) small contractors, each agency would have the other (small contractor) do metadata collection (on each other’s citizens)

This all went away post 911, when each agency could do it officially (whereas before metadata was legally grey)

What was retained, post 911, was the apparatus of grey ness.

Beware trump, with his CIA and NSA versions of his preatorian guards. Now beholden to the new emporer, the old guard may be lynched; Such is the nature of raw naked (super secret) power.

## Gchq spying on trum

“based on the information available to us, we see no indications that Trump Tower was the subject of surveillance by any element of the United States government either before or after Election Day 2016.”

As I recall when deniability is required, that’s when NSA induced gchq to do the spying on Americans (contact with foreign targets, of course)

Trump must be up by now on how and what NSA do!?

He won’t be as adept as Obama. But give him time. The absolute power of secret spying will either tame his dictatorial impulses or get him rapidly impeached.

## Turings on permutations focus on mean = 1

Finally I understand the meaning of the mean being 1.

Id figured a while ago that he was interested in the shuffle around 1.

But now we get that in eigenvector explanation (which is better than the shuffle!)

any distribution is the sum of 1/n of the 1 eigenvector + a superposition of weighted others.

## Constant functions, in Turing op paper

http://www.stat.uchicago.edu/~pmcc/reports/quotient.pdf

Professional version of my similar drivel at https://yorkporc.wordpress.com/2013/10/10/turings-quantum-of-informationa-typex-wheel-wiring-plan-predating-shannons-information-theory/

## https://www.maths.tcd.ie/pub/Maths/Courseware/GroupRepresentations/FiniteGroups.pdf

One d reps of groups , per Turing almost exactly

This is the second paper with modern presentation of two of the argumentation devices Turing used in on permutations. First we finally understand the why of wanting to establish that the math power of d generator is always positive 1. And second, we see why his lemma a is concerned with a power of four.

See paper for other examples of both argumentation devices.

## on permutations and lie algebras

if we reanalyze turings final argument can we say that he is testing for a given representation being one that is a group, given a lie algebra and its generator?

he really says that g has subgroup h which inturn has a centralizer subgroup h1

see http://www.turingarchive.org/viewer/?id=133&title=29

see also k:

### Generalization to Borel sets

This distribution can be generalized to more complicated sets than intervals. If *S* is a Borel set of positive, finite measure, the uniform probability distribution on *S* can be specified by defining the pdf to be zero outside *S* and constantly equal to 1/*K* on *S*, where *K* is the Lebesgue measure of *S*.

from https://en.m.wikipedia.org/wiki/Uniform_distribution_(continuous)

## The PDF for Lie groups and algebras for physicists

https://www.liealgebrasintro.com/publications

let me see two turing arguments

## eigenvalues of random matrices. turings kernel K

https://case.edu/artsci/math/esmeckes/Haar_notes.pdf

i like this writer. she doesnt lose the point with endless symbols.

## lovely Turing mixer theory

## Accumulation Point — from Wolfram MathWorld

http://mathworld.wolfram.com/AccumulationPoint.html

this is the sense of turing- the point at which the measure of diffusion (of colinear differentials in crypto) has become chaotic

recall turing used a doubly transitive operator to mix the plaintext differentials within the output space

## Linear representation theory of quaternion group – Groupprops

https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_quaternion_group

Center and zero

More great Turing

View as CPU design, instruction set for rational – i.e. Probability densities seen in cryptanalysis

View in terms of early comp sci, searching for a computable group (suited to improbability calculus)

## Subgroup structure of quaternion group – Groupprops

https://groupprops.subwiki.org/wiki/Subgroup_structure_of_quaternion_group

Relate center to quotient as Klein.

We saw this in tunny

## Quaternion group picture

http://math.stackexchange.com/questions/1971591/quaternion-group

Contrast with fano plane which holds for octionions

## More Turing simila

Turings on permutation argument

http://math.stackexchange.com/questions/866026/quaternion-group-as-permutation-group

## Eigenvalues and eigenkets in turings on permutations

Looking at turnings overall argument form, I now see that he does address pure states. When he talks about those group element with particular discriminators,these are the eigenkets for his rotational (pre-measurement) operator.

His system is still constrained to be looking at lambda2 (max difference from constants etc).

Interesting also that the inner product norm is to him just a (easy sum of squares) measure of the squiggle path one takes, with respect to another vector (including his eigenket vectors, k).

## Metric and Topological Spaces index

http://www-history.mcs.st-and.ac.uk/~john/MT4522/index.html

excellent uk teaching.

as turing was taught it.

## Fano plane – WikiVisually

http://wikivisually.com/wiki/Fano_plane

Back to the future

Gives a nice notion of distance for measurable probability densities such as those found in Markova ciphers resisting 1943 era differential cryptanalysis

## http://www.ams.org/journals/bull/1931-37-12/S0002-9904-1931-05281-2/S0002-9904-1931-05281-2.pdf

The Turing era teaching on doubly transitive and a sequence of h1….hi, playing the role of n des rounds making a markov cipher